Paragraph: Given two fixed points A and B in a plane and a positive real number k not equal to unity. Introduce a Given two fixed points A and B in a pla choosing mid-point of A B as origin and the positive direction of x axis is chosen from A to B . The locus of points M for which the equation M A M B =k holds is a circle C_k . Question: If the circles C_ k₁ and C_ k₂ (0 k₁ 1,0 k₂ 1 ) , are symmetric to each other w

Step-by-step solution

Let A (-a, 0) and B (a, 0) Locus of point M is (x+a)^2+y^2 (x-a)^2+y^2 =k^2 ( . As . M A M B =k ) C_k x^2+y^2+2 a x (1+k^2 ) (1-k^2 ) +a^2=0 If C_ k₁ and C_ k₂ are symmetric about y -axis ( 1+k₁^2 1-k₁^2 )+ ( 1+k₂^2 1-k₂^2 )=0 k₁ k₂=1 Let any circle passes through A and B Its centre will always lie on y -axis x^2+(y- )^2=a^2+ ^2 Angle between the circles is given by = - ⁻¹ ( r₁^2+r₂^2-d^2 2 r₁ r₂ ) , where r₁ and r₂ are radii of circles and d is distance between their centres = - ⁻¹ [ a^2 ( 1+k^2 1-k^2 )^2-a^2+a^2+